# tangentCone(Ideal)

## Synopsis

• Usage:
tangentCone I
• Function: tangentCone
• Inputs:
• Optional inputs:
• Strategy => default value Local, Local or Global
• Outputs:
• an ideal, the ideal of the tangent cone of the subvariety defined by I at the point defined by the variables of the ring, with a minimal set of generators

## Description

The tangent cone is the ideal that defines gr(R/I), where R is the ring containing I, and gr is the associated graded ring formed with respect to maximal ideal generated by the variables.

The algorithm follows the method of Proposition 15.28 in the book Commutative Algebra with a View Toward Algebraic Geometry by David Eisenbud (Springer, Graduate Texts in Mathematics, volume 150).

 ```i1 : R = QQ[x,y,z] o1 = R o1 : PolynomialRing``` ```i2 : tangentCone ideal "xz-y3,yz-x4,z2-x3y2" 2 4 o2 = ideal (z , y*z, x*z, y ) o2 : Ideal of R``` ```i3 : tangentCone ideal "z2-x5,zx-y3" 2 3 6 o3 = ideal (z , x*z, y z, y ) o3 : Ideal of R``` ```i4 : tangentCone ideal "x3+x2z2,x2y+xz3+z5" 2 3 2 3 5 6 7 9 o4 = ideal (x y, x , x z , 2x*y*z - x*z , x*z , y*z ) o4 : Ideal of R``` ```i5 : betti oo 0 1 o5 = total: 1 6 0: 1 . 1: . . 2: . 2 3: . . 4: . 1 5: . . 6: . 1 7: . 1 8: . . 9: . 1 o5 : BettiTally```